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CorollaryStatement: Literature-sourcedProof: Literature-sourcedSession-authored (Fable 5 assisted)precheck passjudge pass (gpt-5.6-terra)audited 2026-08-27
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The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

  • Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
  • AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
  • AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

A ring is reduced exactly when zero is an intersection of primes

Statement

Assume the Axiom of Choice.

A commutative ring R is reduced if and only if its zero ideal is the intersection of its prime ideals.

Facts & Assumptions

Given: A commutative ring R and the Axiom of Choice.

[L1]

R is reduced exactly when Nil⁡(R)=(0) (The nilradical and reduced rings).

[L2]

The nilradical is the intersection of all prime ideals (The nilradical is the intersection of all prime ideals).

Proof

technique · direct
1.1

If R is reduced, then Nil⁡(R)=(0) by [L1]. Applying [L2] yields (0)=⋂p∈Spec⁡Rp.

L1L2
1.2

Conversely, if (0)=⋂p∈Spec⁡Rp, then [L2] shows that Nil⁡(R)=(0). Now [L1] gives that R is reduced.

L1L2
2.1

Steps 1.1 and 1.2 prove the equivalence.

step 1.1step 1.2∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

4 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources